It probably depends as much on the rider's ability and confidence.
There are some horses that may have never been on trail before, but having a competent rider gives the horse the confidence in new surroundings or uncertain circumstances. Even if the horse balks or acts up, the competent rider will be able to handle the situation properly and end with a positive result.
But you could take that same horse paired up with a rider who is really tense about being out on trail, and that's a potential disaster in the making. The rider will likely become frustrated, the horse may have such an unpleasant experience that it will be even more difficult to ride on trail in the future.
If at all possible, find someone who has a seasoned, BTDT trail horse to ride with. Having another horse along, especially one that takes on hills, ditches, streams, mud, and wildlife without a second thought, will help to build the confidence of the inexperienced trail horse. Along with that, the other rider can help you, offer suggestions or other tips, and be another set of eyes and ears for things to be on the lookout for (loose footing, low branches, etc).
Don't try to do too much in your first ride, especially if the horse has only ever been ridden in an arena or if the area where you plan on riding has a lot of hills. Trail riding is a fantastic physical and mental exercise for horses, but you don't want to wear him out.